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On some quasianalytic classes of $C^\infty$ functions

Abdelhafed Elkhadiri

Abstract

This expository article is devoted to the notion of quasianalytic classes and the Borel mapping. Although quasianalytic classes are well known in analysis since several decades. We are interested in certain properties of Denjoy-Carleman's quasianalytic classes, such as the non-surjectivity of the Borel mapping, the property of monotonicity. We try to see if it remains true for other quasianalytic classes, such as for example, the classes of indefinitely differentiable functions definable in a polynomially bounded o-minimal structures. What motivated this is the fact of having shown in a previous article the existence of quasianalytic classes where Borel mapping is surjective.

On some quasianalytic classes of $C^\infty$ functions

Abstract

This expository article is devoted to the notion of quasianalytic classes and the Borel mapping. Although quasianalytic classes are well known in analysis since several decades. We are interested in certain properties of Denjoy-Carleman's quasianalytic classes, such as the non-surjectivity of the Borel mapping, the property of monotonicity. We try to see if it remains true for other quasianalytic classes, such as for example, the classes of indefinitely differentiable functions definable in a polynomially bounded o-minimal structures. What motivated this is the fact of having shown in a previous article the existence of quasianalytic classes where Borel mapping is surjective.
Paper Structure (12 sections, 27 theorems, 145 equations)

This paper contains 12 sections, 27 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Quasi-analytic functions of real variable.
  3. Quasi-analytic functions according to Denjoy’s point of view.
  4. Some machinery from the theory of metric space.
  5. Monotonicity property for quasianalytic Denjoy-Carleman class
  6. Borel mapping
  7. Quasi-analytic classes associated to a sequence of integers
  8. Monotonicity property for quasianalytic classes associated to a sequence of integers
  9. The Abel-Gontcharoff polynomials
  10. Some estimate of Abel-Gontcharoff polynomial
  11. Monotonicity property for some quasianalytic class
  12. appendix

Key Result

Theorem 2.1

Let $f$ be a $C^\infty$ function on the interval $[a,b]$. The function $f$ is completely determined in the whole interval $[a,b]$ by its value and the values of its derivatives in any point of $[a,b]$, if the series of positive terms: is divergent.

Theorems & Definitions (57)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 47 more